KAM theory in configuration space. (English) Zbl 0682.58014

Salamon, Dietmar; Zehnder, Eduard

Comment. Math. Helv. 64, No. 1, 84-132 (1989).

A Hamiltonian system is considered in the 2n-dimensional space and the Hamiltonian is supposed to be periodic in the first n coordinates, i.e. the system is defined on \(T^ n\times R^ n\) where \(T^ n\) is the n- dimensional torus. The problem is transformed into a variational problem whose Euler-Lagrange equation gives rise to a nonlinear partial differential equation. The latter’s solution is a diffeomorphism on \(T^ n\). This solution gives rise to invariant tori of prescribed periods. The proofs are intricate. The existence proof is based upon a Newton type iteration technique. For given prescribed periods the uniqueness of the corresponding invariant torus is also proved.

Reviewer: M.Farkas

MSC:

37J99	Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37C80	Symmetries, equivariant dynamical systems (MSC2010)

Keywords:

variational problem; nonlinear partial differential equation; diffeomorphism; iteration technique; invariant torus